I am currently pondering over the following question:

For a field $K$, denote by $K[X]$ and $K[[X]]$ the ring of multivariate polynomials and the ring of multivariate power series, respectively. Given $p \in K[X]$ and $s \in K[[X]]$, under what conditions is the product $p \cdot s$ again a polynomial, i.e., an element in $K[[X]]$?

Clearly, if $p = s^{-1}$, then $p \cdot s = 1 \in K[X]$. More generally, if $p$ and $s$ can be factored as $p = qt^{-1}$ and $s = rt$, with $q,r \in K[X]$ and $t \in K[[X]]$ invertible (s.t. $t^{-1}$ is a polynomial), then $p \cdot s = q r \in K[X]$.

Now, my suspicion would be that $p\cdot s$ is a polynomial if and only if we are in the situation illustrated above, where some $t$ cancels with $t^{-1}$. However, so far, I was not able to prove this (nor to find a counterexample).

Maybe someone can help me by either finding a proof, providing a different characterisation, or a counterexample. Any help is highly appreciated!

I am currently pondering over the following question:

For a field $K$, denote by $K[X]$ and $K[[X]]$ the ring of multivariate polynomials and the ring of multivariate power series, respectively. Given $p \in K[X]$ and $s \in K[[X]]$, under what conditions is the product $p \cdot s$ again a polynomial, i.e., an element in $K[X]$?

Clearly, if $p = s^{-1}$, then $p \cdot s = 1 \in K[X]$. More generally, if $p$ and $s$ can be factored as $p = qt^{-1}$ and $s = rt$, with $q,r \in K[X]$ and $t \in K[[X]]$ invertible (s.t. $t^{-1}$ is a polynomial), then $p \cdot s = q r \in K[X]$.

Now, my suspicion would be that $p\cdot s$ is a polynomial if and only if we are in the situation illustrated above, where some $t$ cancels with $t^{-1}$. However, so far, I was not able to prove this (nor to find a counterexample).

Maybe someone can help me by either finding a proof, providing a different characterisation, or a counterexample. Any help is highly appreciated!